MATH 140 5.5 Exercise 1
Finding the domain of a piecewise function and evaluating it
Problem: Find area of triangle with side lengths 100, 150, and 200 cm.
Python Code:
from sympy import *
#Find area of triangle with side lengths 100, 150, and 200 cm
a=100
b=150
c=200
s=(a+b+c)/2
# Area = square root of s(s-a)(s-b)(s-c)
Area=sqrt(s*(s-a)*(s-b)*(s-c))
print('The area of the triangle is', Area, 'square centimeters')
Open Python Notebook File
Finding the domain of a piecewise function and evaluating it
Finding the domain of a function that is a fraction containing exponentials and a square root
Determining where a function with a fraction, square root, and logarithm is continuous
Determining where a piecewise function is continuous algebraically
Using the limit definition to find the derivative of a square root function
Finding the derivative of a function with a square root and exponential using the Product Rule
Using derivatives to find the velocity and acceleration functions from the position function
Simplifying an expression with radicals
Rationalizing the denominator of a fraction with square roots
Rationalizing the denominator of a fraction with square roots
Calculating the difference quotient for function with a square root
Finding the domains of functions
Finding the domains of expressions
Review of limits and derivatives of inverse trigonometric functions
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Related rates problems, differentials, linear and quadratic approximations
Review of limits and derivatives of inverse trigonometric functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Derivatives of exponential and logarithmic functions and the exponential model
Derivatives of exponential and logarithmic functions and the exponential model
Riemann sums, including approximating areas under curves.
Using Reimann sums and the Fundamental Theorem of Calculus
Using Riemann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Finding the derivative of a function using the limit definition
Using a linear approximation to approximate the value of a function
Review of work and average value
Reviewing u-Substitution
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Converting parametric equations into a Cartesian equation and graphing
Finding a Taylor Polynomial for a function
Finding the area of a region enclosed by two curves
Finding the area of a region bounded by curves
Finding the volume of a solid of revolution using the washer method
Finding the volume of a solid of revolution using cylindrical shells
Integrating a function with a square root using u-substitution
Using the comparison test to determine if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Related rates problems, differentials, linear and quadratic approximations
Review of limits and derivatives of inverse trigonometric functions
Proving a property of scalar multiplication for limits using the epsilon-delta definition and using the Squeeze Theorem for Limits.
Approximation and Newton's Method, and limits and derivatives of exponential functions
Using the limit definition to find derivatives of functions and vector functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Derivatives of exponential and logarithmic functions and the exponential model
Riemann sums, including approximating areas under curves.
Using Reimann sums and the Fundamental Theorem of Calculus
The definition and properties of definite integrals
Using Riemann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Review of work and average value
Reviewing u-Substitution
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Finding a Taylor Polynomial for a function
Using integrals to solve work problems and find the average value
Determining the domain of a three-dimensional vector function
Finding a vector equation for the tangent line to a three-dimensional vector function
Finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Explaining the Pythagorean Theorem and using it to find a missing side in a right triangle
Finding the roots of a square root function
Finding the domain of a function with a fraction and radicals
Finding the domain of a function with a quadratic inside a square root
Finding the composition of two functions with a square root and a fractional exponent
Finding the inverse of a square root function and its domain
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
An introduction to installing and getting started with Python
Performing basic numerical calculations in Python
Plotting an expression in Python and finding a numerical approximation for the solution
Graphing a piecewise function using Python
Solving equations symbolically in Python and interpreting the results
Using Python to plot a parametrized curve and its tangent line at a point
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
Using Python to simplify a difference quotient and find a derivative using the limit definition
Using Python to find the tangent line to a parametric equation and plot the two graphs
Using Python to numerically estimate a limit, graphically estimate a limit, and find the exact limit
Using Python's list comprehension tool to find several higher order derivatives of a function at once
Using Python to approximate a definite integral using left endpoint Riemann sums
Solving definite and indefinite integrals in Python
Using Python to find the absolute maximum and minimum of a function on a closed interval
Using Python to find the equation of the tangent line to a curve and graphing the result
Finding derivatives in Python and solving for the rate of change of a force
Finding when the tangent line to a function is horizontal in Python
Using Python to find the intervals where a function is increasing or decreasing
Solving a multistep word problem in Python and graphing the resulting function
Finding the volume of a solid of revolution in Python
Using for loops in Python to generate a recursively defined sequence
Using Python to calculate the partial sums of a series to estimate the value of the series
Using Python to numerically, graphically, and analytically find the limit of a sequence
Using Python to identify the region between curves and then evaluate the area of the region
Using Python to calculate the amount of work done in moving a spring using Hooke's Law
Using Python to calculate the work done in pumping the liquid from a tank whose ends are semicircles
Video 29 in a series of Python instructional videos
Using Python to find the radius and interval of convergence of a power series with the Ratio Test
Explaining domain restrictions for denominators, even roots, and logarithms
Finding the domain of even and odd roots
Finding the domain of a function with a denominator, square root, and logarithm
Finding the domain of a function with a square root in the denominator
Solving an equation that contains rational expressions
Solving an equation that contains rational expressions
Explaining the general properties of even and odd root functions
How to rewrite radical terms as power terms
Explaining how to solve radical equations and then solving example problems
Solving equations with even roots and checking the solutions
Solving equations with multiple radicals
Review of limits and derivatives of inverse trigonometric functions